You must arrange three mystical stones in a line. They have masses 10.0 kg, 7.00 kg,and 3.00 kg. You place the 7.00 kg and 3.00 kg stones 5.00 m apart. Where do you need to place the 10.0 kg stone for the center of mass of the three stones to be 2.25 m from the 7.00 kg stone and 2.75 m from the 3.00 kg stone?

To determine the position where you need to place the 10.0 kg stone, we can use the concept of center of mass. The center of mass of a system of objects is the weighted average position of all the individual masses.

In this case, we have three stones with masses of 10.0 kg, 7.00 kg, and 3.00 kg. Let's denote the position of the 7.00 kg stone as x and the position of the 3.00 kg stone as y. We need to find the value of x.

From the given information, we know that the center of mass is located 2.25 m from the 7.00 kg stone and 2.75 m from the 3.00 kg stone.

To find x, we can use the following equation for the center of mass of a system of two objects:

m1x1 + m2x2 = (m1 + m2) * x_cm

where m1 and m2 are the masses of the two objects, x1 and x2 are their respective positions, and x_cm is the position of the center of mass.

Substituting the known values, we have:

(7.00 kg) * x + (3.00 kg) * 5.00 m = (7.00 kg + 3.00 kg) * 2.25 m

Simplifying the equation:

7.00x + 15.00 = 10.00 * 2.25

7.00x = 22.50 - 15.00

7.00x = 7.50

Dividing both sides of the equation by 7.00:

x = 7.50 / 7.00

x ≈ 1.0714

Therefore, you need to place the 10.0 kg stone at approximately 1.0714 m from the 7.00 kg stone to achieve a center of mass located 2.25 m from the 7.00 kg stone and 2.75 m from the 3.00 kg stone.